The main reason ammonia products are used in a pool is that they form monochloramine that doesn't bind to CYA so in a pool with high CYA levels it's a way of "cheating" to get a higher active oxidizer level. This is especially true against algae that tend to look at monochloramine as if it were ammonia which is something they can use as a source of nitrogen, but the monochloramine kills them. However, your pool doesn't have very high CYA so shocking with chlorine should be quite effective. The key for black algae is to scrape off the waxy surface layer (the "heads") so that the chlorine can kill the algae below. Some people just maintain their pools at a somewhat higher FC level and brush regularly and the algae will slowly get killed while others go to shock level to try and get it over with more quickly -- up to you. At Ben's recommended FC/CYA levels, neither green nor black algae will be able to grow (yellow/mustard algae requires higher levels so is usually best to get rid of completely, if possible).